15,872 research outputs found
Increasing the power of the verifier in Quantum Zero Knowledge
In quantum zero knowledge, the assumption was made that the verifier is only
using unitary operations. Under this assumption, many nice properties have been
shown about quantum zero knowledge, including the fact that Honest-Verifier
Quantum Statistical Zero Knowledge (HVQSZK) is equal to Cheating-Verifier
Quantum Statistical Zero Knowledge (QSZK) (see [Wat02,Wat06]).
In this paper, we study what happens when we allow an honest verifier to flip
some coins in addition to using unitary operations. Flipping a coin is a
non-unitary operation but doesn't seem at first to enhance the cheating
possibilities of the verifier since a classical honest verifier can flip coins.
In this setting, we show an unexpected result: any classical Interactive Proof
has an Honest-Verifier Quantum Statistical Zero Knowledge proof with coins.
Note that in the classical case, honest verifier SZK is no more powerful than
SZK and hence it is not believed to contain even NP. On the other hand, in the
case of cheating verifiers, we show that Quantum Statistical Zero Knowledge
where the verifier applies any non-unitary operation is equal to Quantum
Zero-Knowledge where the verifier uses only unitaries.
One can think of our results in two complementary ways. If we would like to
use the honest verifier model as a means to study the general model by taking
advantage of their equivalence, then it is imperative to use the unitary
definition without coins, since with the general one this equivalence is most
probably not true. On the other hand, if we would like to use quantum zero
knowledge protocols in a cryptographic scenario where the honest-but-curious
model is sufficient, then adding the unitary constraint severely decreases the
power of quantum zero knowledge protocols.Comment: 17 pages, 0 figures, to appear in FSTTCS'0
Quantum Proofs
Quantum information and computation provide a fascinating twist on the notion
of proofs in computational complexity theory. For instance, one may consider a
quantum computational analogue of the complexity class \class{NP}, known as
QMA, in which a quantum state plays the role of a proof (also called a
certificate or witness), and is checked by a polynomial-time quantum
computation. For some problems, the fact that a quantum proof state could be a
superposition over exponentially many classical states appears to offer
computational advantages over classical proof strings. In the interactive proof
system setting, one may consider a verifier and one or more provers that
exchange and process quantum information rather than classical information
during an interaction for a given input string, giving rise to quantum
complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum
analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit
some properties from their classical counterparts, they also possess distinct
and uniquely quantum features that lead to an interesting landscape of
complexity classes based on variants of this model.
In this survey we provide an overview of many of the known results concerning
quantum proofs, computational models based on this concept, and properties of
the complexity classes they define. In particular, we discuss non-interactive
proofs and the complexity class QMA, single-prover quantum interactive proof
systems and the complexity class QIP, statistical zero-knowledge quantum
interactive proof systems and the complexity class \class{QSZK}, and
multiprover interactive proof systems and the complexity classes QMIP, QMIP*,
and MIP*.Comment: Survey published by NOW publisher
The Random Oracle Methodology, Revisited
We take a critical look at the relationship between the security of
cryptographic schemes in the Random Oracle Model, and the security of the
schemes that result from implementing the random oracle by so called
"cryptographic hash functions". The main result of this paper is a negative
one: There exist signature and encryption schemes that are secure in the Random
Oracle Model, but for which any implementation of the random oracle results in
insecure schemes.
In the process of devising the above schemes, we consider possible
definitions for the notion of a "good implementation" of a random oracle,
pointing out limitations and challenges.Comment: 31 page
Superresolution without Separation
This paper provides a theoretical analysis of diffraction-limited
superresolution, demonstrating that arbitrarily close point sources can be
resolved in ideal situations. Precisely, we assume that the incoming signal is
a linear combination of M shifted copies of a known waveform with unknown
shifts and amplitudes, and one only observes a finite collection of evaluations
of this signal. We characterize properties of the base waveform such that the
exact translations and amplitudes can be recovered from 2M + 1 observations.
This recovery is achieved by solving a a weighted version of basis pursuit over
a continuous dictionary. Our methods combine classical polynomial interpolation
techniques with contemporary tools from compressed sensing.Comment: 23 pages, 8 figure
Order-Revealing Encryption and the Hardness of Private Learning
An order-revealing encryption scheme gives a public procedure by which two
ciphertexts can be compared to reveal the ordering of their underlying
plaintexts. We show how to use order-revealing encryption to separate
computationally efficient PAC learning from efficient -differentially private PAC learning. That is, we construct a concept
class that is efficiently PAC learnable, but for which every efficient learner
fails to be differentially private. This answers a question of Kasiviswanathan
et al. (FOCS '08, SIAM J. Comput. '11).
To prove our result, we give a generic transformation from an order-revealing
encryption scheme into one with strongly correct comparison, which enables the
consistent comparison of ciphertexts that are not obtained as the valid
encryption of any message. We believe this construction may be of independent
interest.Comment: 28 page
Non-Cooperative Rational Interactive Proofs
Interactive-proof games model the scenario where an honest party interacts with powerful but strategic provers, to elicit from them the correct answer to a computational question. Interactive proofs are increasingly used as a framework to design protocols for computation outsourcing.
Existing interactive-proof games largely fall into two categories: either as games of cooperation such as multi-prover interactive proofs and cooperative rational proofs, where the provers work together as a team; or as games of conflict such as refereed games, where the provers directly compete with each other in a zero-sum game. Neither of these extremes truly capture the strategic nature of service providers in outsourcing applications. How to design and analyze non-cooperative interactive proofs is an important open problem.
In this paper, we introduce a mechanism-design approach to define a multi-prover interactive-proof model in which the provers are rational and non-cooperative - they act to maximize their expected utility given others\u27 strategies. We define a strong notion of backwards induction as our solution concept to analyze the resulting extensive-form game with imperfect information.
We fully characterize the complexity of our proof system under different utility gap guarantees. (At a high level, a utility gap of u means that the protocol is robust against provers that may not care about a utility loss of 1/u.) We show, for example, that the power of non-cooperative rational interactive proofs with a polynomial utility gap is exactly equal to the complexity class P^{NEXP}
Adiabatic Quantum State Generation and Statistical Zero Knowledge
The design of new quantum algorithms has proven to be an extremely difficult
task. This paper considers a different approach to the problem, by studying the
problem of 'quantum state generation'. This approach provides intriguing links
between many different areas: quantum computation, adiabatic evolution,
analysis of spectral gaps and groundstates of Hamiltonians, rapidly mixing
Markov chains, the complexity class statistical zero knowledge, quantum random
walks, and more.
We first show that many natural candidates for quantum algorithms can be cast
as a state generation problem. We define a paradigm for state generation,
called 'adiabatic state generation' and develop tools for adiabatic state
generation which include methods for implementing very general Hamiltonians and
ways to guarantee non negligible spectral gaps. We use our tools to prove that
adiabatic state generation is equivalent to state generation in the standard
quantum computing model, and finally we show how to apply our techniques to
generate interesting superpositions related to Markov chains.Comment: 35 pages, two figure
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